Gérard Gagneux , Roland Masson , Anne Plouvier-Debaigt , Guy … · Mathematical Modelling and Numerical Analysis ESAIM: M2AN Modélisation Mathématique et Analyse Numérique

Mathematical Modelling and Numerical Analysis ESAIM: M2AN

Modelisation Mathematique et Analyse Numerique M2AN, Vol. 37, No 2, 2003, pp. 373–388

DOI: 10.1051/m2an:2003032

VERTICAL COMPACTION IN A FAULTED SEDIMENTARY BASIN

Gerard Gagneux1, Roland Masson

2, Anne Plouvier-Debaigt

1, Guy Vallet

1

and Sylvie Wolf2

Abstract. In this paper, we consider a 2D mathematical modelling of the vertical compaction effectin a water saturated sedimentary basin. This model is described by the usual conservation laws, Darcy’slaw, the porosity as a function of the vertical component of the effective stress and the Kozeny-Carmantensor, taking into account fracturation effects. This model leads to study the time discretization of anonlinear system of partial differential equations. The existence is obtained by a fixed-point argument.The uniqueness proof, by Holmgren’s method, leads to work out a linear, strongly coupled, system ofpartial differential equations and boundary conditions.

Mathematics Subject Classification. 35Q35, 76S05, 35J65.

Received: July 10, 2002. Revised: January 22, 2003.

1. Introduction

Extracted of [15]: “The constitution of a sedimentary basin during the geologic history implies processesof: sedimentation, erosion, compaction, eviction and transfer of fluids, thermic transfer and of diagenesis, ofwhich outcome is a geologic structure capable of establishing a reservoir of hydrocarbons or a deposit of mineralresources.” The modelling, at a geological scale, of these various mechanisms and their numerical simulationsestablish a promising tool for the evaluation of the oil potential of basins (see [5, 23] and [24]).

The simulation of the genesis and the migration of hydrocarbons in the sedimentary coverage has to take intoaccount sedimentation and erosion phenomena, and so compaction of sediments. From then on, it is necessaryto consider poromechanical models.

The reader interested in similar problems, outside the framework of hydrocarbons, will be able to consultSciarra et al. [25] who consider a binary mixture where a dilatation of pores is observed under extremal pressure.One can also see the importance of compaction in the dynamics of large ice masses as mentioned by Godertet al. in [13].

In this first approach, we shall suppose that the mechanics of cliffs (sediments) deformation ensues from thevertical compaction. The other phenomena of deformation, such as the gliding of the sedimentary layers forexample, will be untidy, either, supposed known in advance and implicitly contained in the data of the problem.It will be enough then to consider a rheological model allowing to express the porosity variations according to

Keywords and phrases. Porous media, vertical compaction, sedimentary basins, fault lines modelling.

1 Laboratoire de Mathematiques Appliquees, Universite de Pau et des Pays de l’Adour, BP 576, 64012 Pau Cedex, France.e-mail: [email protected], [email protected] Institut Francais du Petrole, 1 et 4 avenue de Bois-Preau, BP 311, 92852 Rueil-Malmaison Cedex, France.e-mail: [email protected]

c© EDP Sciences, SMAI 2003

374 G. GAGNEUX ET AL.

Figure 1. Faulted domain.

the only vertical constraint. This simplifying working hypothesis does not remove anything in the capability ofthe model as it is noticed by Luo et al. [17], Perez [22] or Wangen et al. [26] and [27].

The current study concerns a 2D monophasic model in a faulted porous medium. It takes into accountthe vertical compaction and the fracturing according to the state of the effective constraint, with an effect ofthreshold of release.

There are few mathematical publications on compaction models in sedimentary basins. Most of them concernsnumerical aspects (see for example Badea [2], Fowler et al. [8], Ismail-zade et al. [14], Wangen [26], Wangenet al. [27] or Zakarian et al. [29]).

Our goal is to give some mathematical tools in order to analyse such models. Numerical aspects are actuallystudied at the Institut Francais de Petrole (Schneider et al. [23, 24]).

2. Mathematical modelling

As mentioned in the previous section, the anisotropy is mainly vertical, then a 2D model is considered inthe plane Oxy with origin O, horizontal axis x and vertical axis z pointing to the direction of gravity. In thistheoretical study, we consider that the physical domain Ω is decomposed into three sub-domains (cf. Fig. 1):sub-domains Ω1 and Ω3 represent two parts of a sedimentary basin Ω, separated by a fault Ω2. The thicknessof this fault is small compared with the size of Ω1 and Ω3, but it is not unimportant compared with thephysical phenomena which it generates. So, we shall consider Ω2 as a separate sub-domain with appropriatecharacteristics, i.e. allowing a free passage in compulsory directions or causing obstructions.

This working hypothesis becomes delicate in numerical analysis, where it can be preferable to model thisfault as an interface (so without thickness). From then on, this has to lead to artificial boundary conditions inorder to restore the real role of this fault in the water – sediments traffic. One finds then, in previous studies,different subtleties.

In his approach, [22] imposes a continuance of the stream through the fault and introduces a parameter δto take into account the discontinuity of the pressure between the superior wall and the lower wall. Thisdiscontinuity is necessary, otherwise, it would mean that this fault, with a null Lebesgue measure, is neglectedin the system of equations and so would not exist physically.

As indicated by the author, the problem is that the determination of δ is purely empirical. Furthermore,this internal discontinuity prevents the representative function of the water pressure from being in a Sobolev’sspace of first order on Ω, that does not receive any convincing explanations.

In an other approach, one imposes the continuity of the traces on both sides of the fault. But then, asmentioned above, the system of equations alters the physical reality. Then, in the modelling of the fault,

VERTICAL COMPACTION IN A FAULTED SEDIMENTARY BASIN 375

one has to use a term of order 1, and one may have jumps across the interface. It gives, in the formulation ofequations, a supplementary term: a measure with support in the interface-fault, as if it was a well. Furthermore,it does not seem realistic that the pressure is the same on both sides of the fault.

In our approach, we adopt the principles of continuity of traces and of fluxes to write the relations whichgovern the water pressure in each interface. So, we take into account the physical process in Ω2 and jumps existfrom an extreme edge to the other.

If for numerical motivations, one has to consider that the fault is an interface, a last approach could be basedon an asymptotic analysis with regard to ε (the parameter of the fault thickness) in order to find one of theabove mentioned configurations.

The above proposed method, which represents a simplified model (one fault which cuts in two the studieddomain) can be generalized to a sedimentary basin with several faults and several types of sediments. The mainconsideration is to be able to decompose the domain in a certain number of sub-domains with known geologic,rheologic, ..., characteristics.

In a first part, we are interested in the mathematical analysis of the model, where compactions are of weakamplitude. We study then the time semi-discretisation of the system. The existence of a solution results fromSchauder–Tychonoff’s fixed point theorem, in the separable hilbertian framework. Then, uniqueness is provedwith a technique of transposition, inspired by Antontsev and Domansky’s works [1] on the analytical studyof diphasic filtration system. This technique, generalizing the method of pivot space changing [16], classicallyreturns the study of the uniqueness of the primal problem to the study of the existence for the dual problem.This last one admits at least a solution due to Lax–Milgram’s theorem.

In a second part, we consider a model where compaction is of general amplitude. To study this problem ofpredictive-corrective type, we do not make reference to the narrow-mindedness of certain coefficients in the statelaws. Uniqueness is obtained by applying Fredholm’s alternative, associated to a weak maximum principle.

2.1. Notations

Let α ∈ s, w (w label for water and s label for sediments) and i ∈ 1, 2, 3 (for domain Ωi):−→V α : speed of phase α, ρα : volumic mass, pw : water presure,Patm : atmospheric pressure, Ki : permeability tensor, σz : total constraint,λi

a and λis : coefficients of anisotropy, ki : absolute permeability, σ : effective constraint,

Si0 : specific area of the porous media, hi : state law of fracturation, φi : porosity,

φir , φ

ia, φi

b, σia, σi

b : being characteristic parameters of the deposited sediment nature,gs : sedimentation speed at the bottom of the ocean, g : gravity acceleration, Hw : height of water,Fi = [ρwφi + ρs(1 − φi)]g, t−→B = (0, ρwg).

2.2. Conservation laws

In each sub-domain Ωi, the conservation laws are the following ones:

1) mass conservation of the sediment and water:

∂

∂t(ρi

s(1 − φi)) + div(ρis(1 − φi)

−→V s) = 0, (1)

∂

∂t(ρwφi) + div(ρwφi

−→V w) = 0. (2)

2) momentum conservation (equilibrium equation)

∂σz

∂z= (φiρw + (1 − φi)ρi

s)g = Fi(φi). (3)

One supposes in the sequel that ρs and ρw are constant on each sub-domain.


2.3. Behavior laws

These conservation laws have to be completed by phenomenologic behavior laws. We held:

φi(−→V w −−→

V s) = −Ki (−→∇pw −−→

B ) (Darcy′s law), (4)σ = σz − pw (Terzaghi′s relation), (5)

φi(σ) = φir + φi

a exp[− σ

σia

] + φib exp[− σ

σib

] (Elastoplastic rheological law), (6)

t−→V s = (0, vs) (Vertical compaction hypothesis), (7)

Ki(σ, pw) = ki(φi)(

λis 0

0 λia

)+

(0 00 hi(σ, pw)

)(Permeability law), (8)

where the tensor, expressed in a strate – antistrate base, rests on Kozeny–Carman’s law with ki(φ) = 0,2 φ3i

Si20 (1−φi)2

·It is completed with the consideration of the fracturation, by means of a supplementary term hi, which intervenesin the physical phenomenon as soon as certain critical threshold is reached.

2.4. Description of the domain and of the boundary conditions

• One supposes (geometrical regularity) that there are four Lipschitzian functions γ1, γ2, f1 and f2, such that(cf. Fig. 1):

Ω = (x, z) ∈ R2, α < x < β, γ1(x) < z < γ2(x), Ω1 = (x, z) ∈ Ω, z > f1(x),

Ω2 = (x, z) ∈ Ω, f1(x) > z > f2(x), Ω3 = (x, z) ∈ Ω, z < f2(x),Γ1 = (α, z), γ1(α) < z < γ2(α), Γ2 = (β, z), γ1(β) < z < γ2(β),Σi = (x, z) ∈ R

2, α < x < β, z = γi(x), fi = (x, z) ∈ Ω, z = fi(x), i ∈ 1, 2·

• Conditions on Σ1 : pw = σz = Patm + ρwgHw =notation

PΣ1 , in particular, according to (5), one has

pw = PΣ1 , σ = 0. (9)

In fact, Domain Ω evolves during time because of the erosion and of the sedimentation by gravitation on thefree boundary Σ1 (bottom of the ocean), according to the law: vs = gs where, for example, gs =

−→Qs.−→n

nz,−→Qs

ensuing from the sedimentary load of the ocean, the direction, the intensity of maritime currents...

• Condition on Σ2 :−→Vw.−→n = 0 and, according to (4), one gets

−t−→n K (−−→∇pw −−→

B ) = −φ(σ)vsnz, (10)

where nz represents the vertical part of the normal vector −→n .Furthermore, Domain Ω evolves, a priori, also during time by the motion of crusts. So Σ2 is also a free

boundary. We shall suppose it fixed in this study, in order to seriate the difficulties.

• Conditions on Γ1 and Γ2 (artificial free boundaries):−→Vw.−→n = 0. In particular, according to (4) and by noticing

that t−→n = (+/ − 1, 0),−→Vs.

−→n = 0, so that one has

−t−→n K (−−→∇pw −−→

B ) = 0. (11)


• Conditions in the internal interfaces of the domain fi, i = 1, 2:

σΩi = σΩi+1 , vs,Ωi = vs,Ωi+1 , pw,Ωi = pw,Ωi+1 ,

− t−→n Ki

(−→∇pw,Ωi −−→B

)= − t−→n Ki+1

(−→∇pw,Ωi+1 −−→B

).

(12)

• Initial conditions (for t = 0, there are no sediments):

Σ1 = Σ2, φi = φir + φi

a + φib, σz = pw = Patm + ρwgHw, so σ = 0.

2.5. Presentation of the system of equations

First, we introduce some notations: Considering IΩi the caracteristic function of Domain Ωi, one sets,

K(x, z, σ, p) =∑

i

Ki(σ, p) IΩi (x, z), φ(x, z, σ) =∑

i

φi(x, z, σ) IΩi(x, z), F (x, z, σ) =∑

i

Fi(σ) IΩi (x, z).

It is important to notice that each of these functions is regular on Ωi and that it has a trace on ∂Ωi (thus on Σi,Γi and fi).

A triple preliminary analysis of the partial differential equations introduced by the model, of the behaviorlaws which structure them and of the boundary and interfaces conditions imposed by the experimentation, leadsto the following choice for the main unknown

σ, vs and pw,

and to the following choice for the system of equations on Ω

∂σ

∂z= F (., ., σ) − ∂pw

∂z, (13)

− ∂

∂tφ(., ., σ) +

∂

∂z(1 − φ(., ., σ))vs = 0, (14)

−divK(., ., σ, pw) (−−→∇pw −−→

B ) +∂

∂zvs = 0. (15)

Equation (13) comes from the equilibrium equation (3) and Terzaghi’s relation (5), equation (14) comes fromthe mass conservation of the sediment (1) and equation (15) comes from the mass conservation laws (1) and (2)and Darcy’s law (4).

In the sequel, the analysis concerns the study of a time discretisation, reasonable method for the approachof slow evolution processes.

We do not plan to study the time continuous system. Indeed, contribution of sedimentation and erosioneffects prevent a good control of the evolution of Ω during time, in order to pass easily to the limit. Here, thehypothesis of monotonicity for example of the application t → Ω(t), according to the ideas of [16] (p. 415), isnot justifiable. However, we can propose a correction of Domain Ω at each iteration. Indeed, boundary Σ1 is afree boundary, subjected to the oceanic phenomena of deposits or erosion.

Referring to a model which authorizes only vertical deformations, one transcribes the principle of the con-servation of sedimentary material quantity on the vertical line, over the point x of Σ2, by

z∗(x)∫Σ2(x)

(1 − φ(., ., σ(x, z)) dz =

Σ1(x)∫Σ2(x)

(1 − φ(., ., σ0(x, z)) dz +(−→

Qs.−→n

)x

.h,


where: (−→Qs.

−→n )x represents the contribution or the volumic loss via the ocean, z∗(x) is the new quotation forthe boundary Σ1 at the top of the point x of Σ2, supposed indeformable

φ(., ., σ(x, z)) = φ(., ., σ(x, z)) if z ≤ Σ1(x), φ(., ., 0) if z ≥ Σ1(x).

Function hx (for any x) which gives, for any z∗,∫ z∗

Σ2(x)(1 − φ(., ., σ(x, z)) dz, is increasing over R

+. For any x,

this equation defines z∗(x) in a unique way. So, it supplies the Cartesian equation of the new profile of Σ1.By keeping in memory the value of the maximal constraint punctually reached [22], let us notice that this

process of rectification can be enriched with the irreversibility consideration of the compaction.In the next section, we present the mathematical analysis of the first iteration of (13, 14) and (15) in the

case of an implicit discretisation. Moreover, weak amplitude compaction (i.e. with a small variation of φ and Kwith respect to the unknown σ and pw) is considered.

In Section 4, we present the mathematical analysis of the first iteration of (13, 14) and (15) in the case of apredictive-corrective type discretisation. General amplitude compaction is considered.

3. Mathematical analysis of a first case: weak amplitude compaction

Let h = ∆t be the iteration step and σ0 be the datum of σ at the previous iteration. The proposed discretisedscheme is based on equations (13) and (15) for the unknown σ and pw and on

φ(., ., σ0) − φ(., ., σ)h

+∂

∂z((1 − φ(., ., σ))vs) = 0, (16)

for the unknown vs.

3.1. Notations of functional analysis

Then, one has to look for the solutions in an adapted cartesian product of first order Sobolev’s spaces. Inorder to do so, one denotes

V f = u ∈ H1(Ω), u|Σ1 = f, (17)

where f is given on Σ1, regular enough so that V f is not empty.

W = u ∈ L2(Ω),∂u

∂z∈ L2(Ω) = L2(α < x < β, H1[γ1(x), γ2(x)]). (18)

W is a separable Hilbert space for its natural norm

∀u ∈ W, ||u||2W =∫

Ω

u2 dx +∫

Ω

(∂u

∂z

)2

dx.

It is provided with a trace operator, linear and continuous for the natural topologies

γ : W → L2(Σ1) × L2(Σ2), u → (u|Σ1 , u|Σ2),

so that the following notation is coherent: W g = u ∈ W, u|Σ1 = g.Furthermore, for any Lipschitzian function f from R into R, the following chain rule holds for the weak

derivatives: ∀u ∈ W , f(u) ∈ W and∂f(u)

∂z= f ′(u)

∂u

∂za.e. in Ω,

where f ′ indicates a bounded Borelian representative of f derivative (it exists in the classic sense almosteverywhere according to Rademacher’s theorem).


3.2. Hypothesis

One assumes that: ∃c, C, M > 0, 0 < c < C < 1,

c ≤ φi ≤ C, 0 ≤ h ≤ M (19)

∀σ1, σ2 ∈ R, σ1 < σ2 ⇒ 0 ≤ φ(., ., σ2) − φ(., ., σ1) ≤ M(σ2 − σ1) (20)

∀σ1, σ2, p1, p2 ∈ R, |hi(σ1, p1) − hi(σ2, p2)| ≤ M(|σ2 − σ1| + |p2 − p1|), i = 1, 2, 3. (21)

σ0 ∈ W 0, ∃q > 2, V PΣ1 ∩ W 1,q(Ω) = ∅, gs ∈ L∞(Σ1), W gs = ∅. (22)

3.3. Definition of a solution

Definition 3.1. One calls solution to System (13, 16, 15) for Conditions (9)–(11) and (12), any (σ, vs, pw) inW 0 × W gs × V PΣ1 such that

(13) and (16) are satisfied a.e. in Ω,and, for any ϕ in V 0,∫

Ω

t−→∇ϕK(., ., σ, pw)(−→∇pw −−→

B)

dx +∫Ω

∂vs

∂zϕdx =

∫Σ2

φ(σ)vsnzϕdσ. (23)

One has to remark that (12) is implicitly contained in the fact that the solutions belong to W 0, W gs and V PΣ1 .Indeed, these functions possess the property of traces countinuity through interfaces fi. It is also contained inthe formulation (23) for the property of flux countinuity.

3.4. Existence of a solution

Proposition 3.2. There exists at least one solution to the problem in the sense of Definition 3.1.

In order to establish the existence of a solution, we suggest to use Schauder–Tikhonov’s fixed point theorem,in the context of hilbertian separable spaces [10]. Therefore, one fixes pw in V PΣ1 . Thus, integration of equa-tions (13) and (16) gives a solution σ(pw) in W 0 and a solution vs(pw) in W gs . It is enough then to inject thesetwo solutions in the following paralinearised version of equation (23)∫

Ω

t−→∇ϕK(., ., σ(pw), pw)(−→∇pw − −→

B)

dx +∫Ω

∂vs(pw)∂z

ϕdx =∫Σ2

φ(., ., σ)vs(pw)nzϕdσ, (24)

to obtain a unique solution pw in V PΣ1 and so, to build an application S from V PΣ1 into himself, defined forany pw by S(pw) = pw.

It is immediate to notice that the solutions of System (13, 16, 15), for the boundary conditions (9)–(11) andthe interface condition (12), are the fixed points of S.

One shows then that S keeps a non empty bounded closed convex set of H1(Ω). It relies mainly on the factthat vs is bounded in L∞(Ω). From then on, with the help of a priori estimations, of subsequences extractionand of weak compactness arguments, one shows that S is weak-sequentially continuous. So, there exists a fixedpoint. The reader interested in technical details of these results is referred to [11].

3.5. Lp regularity of the obtained solution: an extension of N.G. Meyers’s principle

For many coupled non linear systems, the study of solution uniqueness passes by the treatment of trilinearterm integration. From then on, it is necessary to improve the regularity knowledge of the obtained solution.This wellknown result, relaying on a disturbance of the Poisson equation, was initially introduces by [18]. Usedagain by [4, 19] and recently by [20] and [9] within the framework of diphasic models. We propose in [11] anadapted version to our problem, namely, an elliptic equation in a Lipschitzian domain, and of which a corollary is


Proposition 3.3. There exists p0 > 2, k0 > 0, independent of solutions σ and vs so that: pw ∈ W 1,p0 (Ω) and||pw||W 1,p0 ≤ k0.

3.6. Uniqueness of the solution

3.6.1. Presentation of the duality method

Suggested by S.N. Antontsev’s works [1] concerning the analytical study of diphasic filtration systems. Theproposed method of uniqueness is based on a technique of transposition. It is inspired also by Holmgren’sduality method, implemented by Oleınik [21].

This method of duality returns classically the study of the uniqueness of the primal problem to the study ofthe existence of a solution to the dual problem, in compatible functional frameworks.

We denote (σ1, p1, v1) and (σ2, p2, v2) two solutions to system (13, 16, 15) with the boundary conditions (9)–(11) and (12).

After subtracting of equations (13, 16) and (23) verified by (σ1, p1, v1) and (σ2, p2, v2), we shall transport thederivations on the test functions, in order to introduce quantities σ = σ1 − σ2, v = v1 − v2 and p = p1 − p2. Inorder to relieve the demonstration writing, we introduce some notations

φ′ =φ(., ., σ1) − φ(., ., σ2)

σ1 − σ2if σ1 = σ2, φ′ =

∂φ

∂σ(., ., σ1)

on the L2-measurableset σ1 = σ2 ,

F ′ =F (., ., σ1) − F (., ., σ2)

σ1 − σ2if σ1 = σ2, F ′ =

∂F

∂σ(., ., σ1) if σ1 = σ2,

D1K =K(x, z, σ2, p1) − K(x, z, σ2, p2)

pif p = 0, D1K =

∂K(x, z, σ2, p2)∂p

if p = 0,

D2K =K(x, z, σ1, p1) − K(x, z, σ2, p1)

σif σ = 0, D2K =

∂K(x, z, σ1, p1)∂σ

if σ = 0,−→D1(σ2) = t[

−→∇p1 −−→B ] D1K,

−→D2(p1) = t[

−→∇p1 −−→B ] D2K.

Since functions φi, Fi, ki and hi are Lipschitzian, φ′ and F ′ are measurable functions. According to Rademacher’s theorem, one is even able to choose them as bounded Borelian functions. Also,

−→D2(p1) and

−→D1(σ2)

are [L2(Ω)]2 functions. More exactly, thanks to Proposition 3.3, they are elements of [Lp0(Ω)]2.Right now, it is necessary for us to clarify the functional spaces required to look for the solutions to the dual

problem and to legitimize Gauss–Green formulae. For it, one considers

X =

u ∈ L2(Ω), ∀i,∂u

∂z∈ L2(Ωi)

, ||u||2X = ||u||2L2(Ω) +

∑i

∣∣∣∣|du

dz

∣∣∣∣ |2L2(Ωi).

It is a subspace of

BVz =

u ∈ L2(Ω),∂u

∂z∈ Mb(Ω)

(where Mb(Ω) indicates the set of bounded Radon measures on Ω). It has been introduced by [28] for the studyof degenerated hyperbolic-parabolic problems. We suggest to denote by du

dz the absolutely continuous part, withrespect to the Lebesgue measure L, of the measure ∂u

∂z . So, for any u of X ,

du

dz=

∑i

∂u|Ωi

∂zIΩi ,

while∂u

∂z=

du

dzdL +

∑i

(u+ − u−)nz H1 fi ,


where u+ and u− represent the traces of u on the right and on the left of fi (if one chooses the direction givenby −→n , the normal vector to fi, from Ωi to Ωi+1) and H1fi represents the restriction of the 1-dimensionnalHausdorff measure on fi.

Yp =

v ∈ L2(Ω), ∀i,∂v

∂z∈ Lq(Ωi)

, q =

2p

p + 2·

3.6.2. First step: superimposing of the balance states and transposition

Concerning equations verified by v1 and v2: the subtraction of the equations, the integration by parts oneach Ωi, by adding of the i together and by regrouping each terms, it comes, for any α of X ,

∫Ω

σφ′(

v2dα

dz− α

h

)dx +

∫Ω

v (φ (., ., σ1) − 1)dα

dzdx +

∑i

∫fi

σ v2

[φ′+α+ − φ′−α−]

nz dσ

−∑

i

∫fi

v(1 − φ+ (., ., σ1))α+nz dσ +∑

i

∫fi

v(1 − φ− (., ., σ1))α−nz dσ

=∫Σ2

σφ′ v2 α nz dσ −∫Σ2

v(1 − φ (., ., σ1))α nz dσ. (25)

Concerning equations verified by σ1 and σ2: in a same way, one has, for any β of Yp0 (p0 given by Property 3.3),∫Ω

σ

(dβ

dz+ F ′β

)dx +

∫Ω

pdβ

dzdx =

∑i

∫fi

(p + σ) (β+ − β−) ηz dσ +∫Σ2

(p + σ) β ηz dσ. (26)

Concerning the variational equations verified by p1 and p2, a subtraction gives for any γ of V 0,

∫Ω

σ−→D2(p1)

−→∇γ + p−→D1(σ2)

−→∇γ dx +∫Ω

t−→∇γK(., ., σ2, p2)−→∇p dx +

∫Σ2

v γ nz dσ =

∫Ω

v∂γ

∂zdx +

∫Σ2

σφ′v2nzγ dσ +∫Σ2

vφ(., ., σ1)nzγ dσ. (27)

From then on, regrouping equations (25, 26) and (27) allows us to obtain the vectorial variational formulationon X × Yp0 × V 0,∫

Ω

σ

φ′ (v2

dα

dz− α

h) +

dβ

dz

dx +

∫Ω

σF ′β +

−→D2(p1)

−→∇γ

dx −∫Ω

v

(1 − φ (., ., σ1))

dα

dz+

∂γ

∂z

dx

+∫Ω

p

dβ

dz+−→D1(σ2)

−→∇γ

dx +

∫Ω

t−→∇γK(., ., σ2, p2)−→∇p dx

+∑

i

∫fi

σ v2[φ′+α+ − φ′−α−] nz dσ −∑

i

∫fi

σ [β+ − β−] nz dσ

−∑

i

∫fi

v [1 − φ+(., ., σ1)]α+nz dσ +∑

i

∫fi

v [1 − φ−(., ., σ1)]α−nz dσ −∑

i

∫fi

pβ+ − β−

nz dσ

=∫Σ2

σ β + φ′v2(α + γ)nz dσ +∫Σ2

v (1 − φ(., ., σ1))(α + γ) ηz dσ +∫Σ2

pβ ηz dσ. (28)


Thus, it is enough to find (α, β, γ) in X × YP0 × V 0, solution to the formal dual (or transposed) problem in Ω

(1 − φ (., ., σ1))dα

dz+

∂γ

∂z= 0, (29)

φ′(

v2dα

dz− α

h

)+

(dβ

dz+ F ′β

)+−→D2(p1)

−→∇γ = 0, (30)

∂β

∂z+−→D1(σ2)

−→∇γ − Div

K(., ., σ2, p2)−→∇γ

= p, (31)

with the interface condition on fi

[1 − φ+(., ., σ1)]α+ = [1 − φ−(., ., σ1)]α−, β+ − β− = v2[φ′+α+ − φ′−α−], (32)

and the boundary conditions

γ = 0 on Σ1, β = 0, α = −γ on Σ2, − t−→n K(σ2, p2)−→∇γ = 0 on ∂Ω\Σ1. (33)

The dual system (29)–(31) is linear, but it is strongly coupled (by the equations, as well as by the boundaryconditions).

3.6.3. Second step: existence of a solution to the dual problem

We are going to consider here a method by substitution. This method allows us to reduce the dual systemto a linear equation concerning γ, the dual unknown of the pressure, in the framework of the classical Lax–Milgram’s theorem. This theorem, which gives a sufficient condition of existence and uniqueness of a solution,can be too constraining. One can prefer Brezzi–Babuska’s theorem ([7], p. 564 for example). But in our case,because of the coupling, the inf sup conditions do not seem immediate to obtain.

Let γ be fixed in V 0. By a simple quadrature of a linear differential equation on every Ωi (and by using con-ditions on Σ2 and on fi), one obtains the existence and the uniqueness of the solution α[γ] in X to equation (29)with conditions (32) and (33).

Furthermore, the application α : V 0 → X, γ → α[γ] is linear and continuous [11].The same work allows us to obtain a result of existence and uniqueness of β[γ] in Yp0 solution to

dβ

dz[γ] + F ′β[γ] + H [γ] = 0 in Ω where H [γ] = φ′(v2

dα

dz[γ] − α[γ]

h) + D2(p1)∇γ,

β+ − β− = v2[φ′+α+ − φ′−α−] on fi, β[γ] = 0 on Σ2.

One obtains this solution in Yp0 since, ∇p1 ∈ Lp(Ω), ∇γ ∈ [Lp0(Ω)]2 and−→D2(p1)

−→∇γ ∈ Lq0(Ω) where q0 =2p0

p0 + 2·

Also, the application β : V 0 → Yp0 , γ → β[γ] is linear and continuous [11].We are henceforth able to give the dual variational problem and to look for a solution by means of Lax–

Milgram’s theorem. In order to do so, and in agreement with the modelling, we suppose that ||φ′||∞, ||D2K||∞


and ||D1K||∞ are small, in a sense precised in the sequel. We note now

b : V 0 × V 0 −→ R

(u, v) →∫Ω

t−→∇vK(σ2, p2)−→∇u dx +

∫Ω

v−→D1(σ2)

−→∇u dx +∫Ω

vdβ

dz[u] dx +

∑i

∫fi

(β+[u] − β−[u]) v nz dσ.

As β[u] and σ[u] are linear with respect to u, b is a bilinear form.Furthermore, as in dimension 2, the injection of H1(Ω) in Lq′

(Ω) is compact (1q + 1

q′ = 1 ), b is continuouson V 0 × V 0, for its natural topology. Thus, for any u in V 0, if a = nw inf(λa, λs), one has

b(u, u) ≥ a ||−→∇u||2 − c(||φ′||∞ + ||D2K||∞ + ||D1K||∞) ||u||2V 0 ,

and as Poincare’s inequality is valid in V 0, one gets,

∀u ∈ V 0, b(u, u) ≥ a||u||2V 0 − c(||φ′||∞ + ||D2K||∞ + ||D1K||∞)||u||2V 0 .

From then on, hypothesis concerning the smallness of ||φ′||∞, ||D2K||∞ and ||D1K||∞, which finds here itsimplicit formulation, insures the coercitivity of the form b and so, the existence of a solution to the dual system.

In conclusion, there exists (α, β, γ) in X × Yp0 × V 0, verifying the equations (29) and (30) with the condi-tions (32) and (33), and for any ϕ of V 0, the variational equation,

∫Ω

ϕ

[dβ

dz[γ] +

−→D1(σ2)

−→∇γ

]dx +

∫Ω

t−→∇γK(σ2, p2)−→∇ϕ dx +

∑i

∫fi

v2(φ′+α+[u] − φ′−α−[u]) v nz dσ = 〈p, ϕ〉.

In particular, ϕ = p in (28) leads to ||p||2L2(Ω) = 0 and to p1 = p2 a.e.As σ and vs are univocaly obtained for a fixed pw, one has shown the uniqueness of the solution (σ, vs, pw).

3.6.4. Alternatives

Another approach consists in applying Fredholm’s alternative as presented in [6] and [12]. This theorem,concerning the wellknown compatibility conditions of linear systems in finite dimension, becomes widespread inthe case of compact operators. We are going to develop this method of uniqueness in the next section. A thirdone is the method of artificial compressibility where one suggests to replace the elliptic equation

∂β

∂z+−→D1(σ2)

−→∇γ − DivK(σ2, p2)

−→∇γ

= p,

by the parabolic regularization (effect of very weak compressibility)

ε∂γ

∂t+

∂β

∂z+−→D1(σ2)

−→∇γ − DivK(σ2, p2)

−→∇γ

= p.

On the one hand, one needs less hypotheses on the operator (for the existence of a solution) in the paraboliccase than in the elliptic case. On the other hand, one needs a priori estimations, on the obtained solution γε,and needs to pass to the limit on ε, as ε → 0+.

Finally, a last approach consists in using for this system, although linear, a fixed point method, as the oneproposed in the section: existence of a solution.


It consists, for any fixed γ in V 0, in resolving in V 0

∂β

∂z[γ] +

−→D1(σ2)

−→∇θ − DivK(σ2, p2)

−→∇θ

= p,

and in studying the properties of Application S : γ → θ.As the equations are linear, it is the same for S. As the previous study proves that S is continuous, it is

weakly sequentially continuous from V 0 into V 0. So one has to find a non empty bounded closed convex set,invariant by S.

By summing up the study of the second step, it comes at once

||θ||V 0 ≤ c||p|| + c(||φ′||∞ + ||D2K||∞ + ||D1K||∞) ||γ||V 0 ,

so that if η = c(||φ′||∞ + ||D2K||∞ + ||D1K||∞) < 1, the looked for convex is V 0 ∩ B(0, c||p||1−η ).

Even there, hypothesis concerning the smallness of certain coefficients in needed to guarantee that theinvariance of a convex set is gathered.

4. A second case: general amplitude compaction

In this second approach of the discretisation, one does not need any more the smallness of certain coefficients.On the other hand, one considers that φi is a regular function of arguments x, z and σ and that the functionφ : φ(x, z, σ) =

∑i φi(x, z, σ) IΩi (x, z), is continuous.

We propose, for this problem, an approach of predictive-corrective type. That is, if one still denotes by h theiterative step and σ0 the datum of σ at the previous iteration, and if one supposes h large enough1 to neglect∂φ∂t (asymptotically stable system), one looks for (σ, vs, pw) in W 0 × W gs × V PΣ1 solution to

∂σ

∂z= F (., ., σ0) − ∂pw

∂z,

∂(1 − φ(., ., σ))vs

∂z= 0 on Ωi,

−div

K(., ., σ, pw)(−−→∇pw −−→

B)

+∂

∂zvs = 0,

(34)

with the same boundary and interfaces conditions.After an integration of the two first equations, one calls solution to the system any (σ, vs, pw) in W 0×W gs ×

V PΣ1 such that:

σ(x, z) =

z∫Σ1(x)

F (., ., σ0(x, s)) ds − pw(x, z) + PΣ1(x, Σ1(x)), (35)

vs(x, z) =1 − φ(x, Σ1(x), 0)1 − φ(x, z, σ(x, z))

gs(x, Σ1(x)), (36)

∀ϕ ∈ V 0,

∫Ω

t−→∇ϕK(., ., σ, pw)(−→∇pw −−→

B)

dx +∫Ω

∂vs

∂zϕdx =

∫Σ2

φ(., ., σ)vsnzϕdσ. (37)

As for the first case, a fixed point method leads directly to the existence of a solution.

1One works with geologic scale and h may represent a century.


4.1. Uniqueness of the solution

The method used in this section is closed to the one already proposed in Section 3.6.Using the same calculus as in the first step of this duality method, one is led to look for (α, β, γ) a priori

in X × Yp0 × V 0 solution to System

(1 − φ(., ., σ1))dα

dz+

∂γ

∂z= 0,

dβ

dz+ φ′v2

dα

dz+−→D2(p1)

−→∇γ = 0,

∂β

∂z− Div(K(., ., σ2, p2)

−→∇γ) +−→D1(σ2)

−→∇γ = p,

with the interface conditions on fi

[1 − φ+(., ., σ1)]α+ = [1 − φ−(., ., σ1)]α−, β+ − β− = v2[φ′+α+ − φ′−α−],

and the boundary conditions ([3])

γ = 0 on Σ1,

β = 0, α = −γ on Σ2,

−t−→n K(., ., σ2, p2)∇γ = 0 on ∂Ω\Σ1.

But as φ+ = φ− and φ′+ = φ′− ( φ is supposed continuous), the interface conditions on fi become α+ = α−

and β+ = β−.From then on, α and β are looked for in W and an algebraic manipulation leads us to look for γ in V 0

solution to

−Div(K(., ., σ2, p2)

−→∇γ)

+−→D1(σ2)

−→∇γ +φ′v2

1 − φ(., ., σ1)∂γ

∂z−−→

D2(p1)−→∇γ = p,

i.e.−Div(A

−→∇γ) +−→B.

−→∇γ = p,

with the boundary conditions of edge

γ = 0 on Σ1, − t−→n K(., ., σ2, p2)−→∇γ = 0 on ∂Ω\Σ1.

Then, the second step is based on an application of Fredholm’s alternative.To conclude, we consider the following family of bilinear forms (aλ)λ≥0 on V × V

aλ : V × V → R, (u, v) →∫Ω

t−→∇uA−→∇v dx +

∫Ω

−→B.

−→∇u v dx + λ

∫Ω

uv dx,

and a instead of a0. We notice that as−→B ∈ (Lp0(Ω))2, for any u in V ,

−→B.

−→∇u ∈ Lq0(Ω). So thanks tothe continuous injection of V in Lq′

0(Ω), the trilinear term∫Ω

−→B.

−→∇u v dx has a sense for any v of V and aλ iscontinuous. Furthermore, one shows that there is a value of λ such that aλ is V 0-elliptic. So, for any f in L2(Ω),there exists a unique u in V 0 such that

∀v ∈ V, aλ(u, v) =∫

Ω

fv dx. (38)


From then on, a classic application of Fredholm’s alternative [6] allows us to show that the problem: find uin V 0 such that

∀v ∈ V, a(u, v) =∫

Ω

fv dx, (39)

admits a solution, moreover unique, if the trivial solution is the only solution to (39) when f = 0.This last result ensues from the weak maximum principle [12], generalized in Ω ⊂ R

N , as follows:

In a first step, one shows that u is upper bounded. We consider k such that 0 ≤ k <γ = sup essΩ

u (possibly

infinite) and the test function vk = (u − k)+ in (39). It follows the estimation,

c||−→∇vk||2L2(Ω)N ≤ ||−→B||Lp0(Ω)) ||−→∇vk||L2(Ω)N ||vk||Lq′0(Ω),

and, since by construction vk is not null, it comes, for a big enough value of p∗,

||−→∇vk||L2(Ω)N ≤ C ||vk||Lq′0(Ω)≤ C [meas(supp vk)]

p∗−q0p∗q′0 ||vk||Lp∗(Ω),

where C represents an arbitrary constant. Finally, the countinuous injection of V 0 in Lp∗(Ω) leads to

||−→∇vk||L2(Ω)N ≤ C[meas(supp vk)]p∗−q0p∗q′0 ||−→∇vk||L2(Ω)N .

So LN − meas(supp vk) ≥ C > 0, constant independent of k (||−→∇vk||L2(Ω)N > 0). By noticing that supp vk supp vγ when k γ, it ensues from Beppo–Levi’s theorem that

meas(supp vγ) > 0.

So, there is a non negligeable LN -measurable subset Ω0 of Ω, such that u = γ a.e. in Ω0. Moreover, since u inLN -integrable on Ω, γ < +∞.

Second step consists in showing that u is non positive LN -a.e. in Ω. In order to do so, let us consider, forany ε > 0, the test function vε = u+

γ+ε−u+ . From then on, it comes

∫Ω

γ + ε

(γ + ε − u+)2t−−→∇u+ A

−−→∇u+ dx +∫

Ω

−→B.

−−→∇u+ u+

γ + ε − u+dx = 0,

and

c

∫Ω

|−−→∇u+|2(γ + ε − u+)2

dx ≤∫

Ω

|−→B| |−−→∇u+|γ + ε − u+

u+

γ + εdx.

One observes that, 0 ≤ u+

γ+ε ≤ 1. So, if one notes wε = ln[

γ+εγ+ε−u+

], then wε ∈ V 0, the chain rule gives

−−→∇wε =−−−→∇u+

γ+ε−u+ and one gets

c

∫Ω

|−−→∇wε|2 dx ≤∫

Ω

|−→B| |−−→∇wε| dx ≤ ||−→B||L2(Ω) ||−−→∇wε||L2(Ω) N .

In conclusion, by means of Poincare’s inequality, for any ε > 0, ||wε||L2(Ω) ≤ c ||−−→∇wε||L2(Ω)N ≤ C (constantindependent of ε).


If one supposes, by contradiction, the negation of the assertion: u ≤ 0 a.e., γ > 0 and it ensues directly that

(ln[

γ + ε

ε])2

meas(Ω0) =∫

Ω0

(ln[

γ + ε

ε])2

dx

=∫

Ω0

(ln[

γ + ε

γ + ε − u+])2

dx

≤ ||wε||2L2(Ω) ≤ C (independent of ε).

This situation is impossible since(ln

[γ+ε

ε

])2converges towards +∞ if ε converges to 0+. Thus, γ ≤ 0 and by

definition of γ, u ≤ 0 a.e. in Ω.The last step represents the conclusion. It is enough to notice, because of the linear character of the homo-

geneous equation, that if u is a solution, −u is also a solution. Thus u ≤ 0 and u ≥ 0, i.e. u = 0.Thus, f = p in (39) leads to the uniqueness of the solution to the primal problem, as it was presented in the

study of the first case.

5. The case of dimension 3: Control of p0

It is possible to give the same study for a domain of R3 of the shape

Ω =(x, y, z) ∈ R

3, α < x < β, ς < y < τ, γ1(x, y) < z < γ2(x, y)

,

where γi are Lipschitzian functions. Problems are concentrated around the possibility of achieving the techniqueof uniqueness. In particular, the treatment of

∫Ω w

−→D1(σ2)

−→∇γ dx when w ∈ V 0. That is to find again the resultof Proposition 3.3 and more exactly, to know (for technical reasons) if p0 > 3 is possible.

If one resumes the demonstration of this result, such as proposed in [19], one looks, page 46, for a value ofp0 > 2, close to 2, such as, according to the notations of the context: A

1−α(p0)q

λ2−λ1λ2+λ1

n12− 1

p0 ≤ 1.Another method consists, for a fixed value of p0 (p0 > 3 for example), to suppose that λ1 is close to λ2. That

is, the conditioning of the matrice (aij)ij , for the matrix norm subordinate to the euclidian norm, is close to 1.However, in practice, the relative value of coefficients λs and λa which specify Kozeny–Carman’s tensor of

permeability in relation (8) limits the reach of this theoretical remark. The control of p0, according to thespirit of the method, requires that the elliptic problem with symmetric and bounded coefficients is a “small”disturbance of Poisson’s equation. What is not always turned out (case of strong anisotropy). As soon as thestudied environment gives an importance to a space direction (in this particular case, the models of sedimentarybasins enters in this frame via the hypothesis of vertical line compaction), the control of p0 is very checked. Inpractice, p0 remain very close to 2+ (cf. [4]).

Acknowledgements. The authors wish to thank the referee for providing many interesting references in this topic and forvery attentive and careful revision of the paper.

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